When performing some tasks, such as registering your site with a search provider, the following error may occur in your Event Viewer on your search server. This is a known behavior with SQL Server 2012.
Error: SQL database login for 'xxxxxx' on instance 'SQL Server Machine Name' failed. Additional error information from SQL Server is included below.
Login failed for user 'domain\search server$'.
SQL Server 2012 will try to authenticate using the Network Service/local system account typically if your connectionstring has integrated security set to true in your site's web.config.
Another reason is that the application pool for the site may be running under Network Service and not a domain account.
Typically this error is resolved by running the below script on the database referenced in the error, after backing it up:
CREATE LOGIN [domain\machine name$] FROM WINDOWS;